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Unformatted text preview: ARE211, Fall 2007 LECTURE #8: TUE, SEP 25, 2007 PRINT DATE: OCTOBER 10, 2007 (GRAPHICAL2) Contents 2. Graphical Overview of Optimization Theory (cont) 1 2.4. Separating Hyperplanes 1 2.5. Constrained Maximization: One Variable. 3 2.5. Unconstrained Maximization: Several Variables. 5 2.6. Introduction to Taylor’s theorem 14 2.7. Level Sets, upper and lower contour sets and Gradient vectors 17 2. Graphical Overview of Optimization Theory (cont) 2.4. Separating Hyperplanes A very important property of convex sets is that if they are “almost” disjoint—more precisely, the intersection of their interiors is empty—then they can be separated by hyperplanes. For now, we won’t be technical about what a hyperplane is. It’s enough to know that lines, planes and their higherdimensional equivalents are all hyperplanes. As Fig. 4 illustrates, this property is completely intuitive in two dimensions. • In the top two panels, we have convex sets whose interiors do not intersect, and we can put a line between them. Note that it doesn’t matter if the sets are disjoint or not. However, – if the sets are disjoint, then there will be many hyperplanes that separate them. 1 2 LECTURE #8: TUE, SEP 25, 2007 PRINT DATE: OCTOBER 10, 2007 (GRAPHICAL2) DISJOINT INTERIORS HAVE EMPTY INTERSECTION INTERIORS INTERSECT ONE SET ISN’T CONVEX Figure 4. Convex sets whose intersection is empty can be separated by a hyperplane – if they are not disjoint, but their interiors do not intersect, then there may be a unique hyperplane separating them. What condition guarantees a unique hyperplane? Differentiability of the boundaries of the sets. • In the bottom two panels, we can’t separate the sets by a hyperplane. In the bottom left panel, it’s because the sets have common interior points; in the bottom right it’s because one set isn’t convex. Why do we care about them? They crop up all over the place in economics, econometrics and finance. Two examples will suffice here ARE211, Fall 2007 3 (1) The budget set and the upper contour set of a utility function (2) The Edgeworth box Telling you any more would mean teaching you economics, and I don’t want to do that!! 2.5. Constrained Maximization: One Variable. Question: So far, we’ve been declaring that a necessary condition for a local maximum at x * is that f prime ( x * ) = 0. That is, if the slope isn’t zero at x * then I know I don’t have a local maximum at x * . Now that was true, because of the way in which I defined the function f , but it was only true given this caveat. What was the critical part of the definition of f ? In other words, under what conditions can a differentiable function f have a local maximum at x * but the slope isn’t zero at x * ?...
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